Dispersion Exercises


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This page presents multiple data sets and for each you are to find, using RStudio, various measures of dispersion, the range, the quartiles, and the standard deviation. The standard deviation is to be computed twice, once for a sample, and once for a population. While you are at it, you asked to also find the mean, the median, and the mode, which are measures of central tendency. In order that you can check your work, the answers are given for each data set.

Case 1:

For the data set in Table I, find the mean, the median, the mode, the first and third quartiles, and the standard deviation, both treating the data set as a sample and as a population.
Table I
31 25 46 39 39 40 27 33 26 28 35 34 35 23 24 26 38 24 23 25
48 35 26 24 23 27 33 39 30 23 24 45 23 23 24 27 48 27 27 28
23 37 24 45 33 38 24 24 44           
You can generate the values shown in the table via the command:
gnrnd4( 399484802, 2500023) in RStudio.

Answers:
   mean = 30.95918
   median = 27.0
   mode = 24
   range: 23 to 48
    Q1 = 24
    Q3 = 37
    σ = 7.72739 (for population)
    sx = 7.80747 (for sample)


Case 2:

For the data set in Table II, find the mean, the median, and the mode, the first and third quartiles, and the standard deviation, both treating the data set as a sample and as a population.
Table II
245 260 244 258 266 252 268 250 269 250 269 257 243 256 263 263 261 268 252 258
253 264 255 247 260 259 263 264 244 247 265 257 254 249 268 262 260 254 256 262
256 245 246 268 249 262 269 268 252 253 269 260 255 256 256 267 268 262 249 263
264 251 263 244                
You can generate the values shown in the table via the command:
gnrnd4( 450156301, 2700243) in RStudio.

Answers:
   mean = 257.65625
   median = 258.0
   mode = 268

   range: 243 to 269
    Q1 = 252
    Q3 = 263.25
    σ = 7.68566 (for population)
    sx = 7.74641 (for sample)


Case 3:

For the data set in Table III, find the mean, the median, and the mode, the first and third quartiles, and the standard deviation, both treating the data set as a sample and as a population.
Table III
244 251 271 254 272 236 259 263 247 266 269 269 247 273 240 254 229 255 243 264
264 271 230 261 268 273 238 259 240 263 265 246 267 269 231 244 265 267 236 257
248 267 262 271 273 230 273 262 268 264 246 271 238 248 252 258 243 272 253 273
262 247 267 270 256 262 263 255 247 272 253 250 262 240 256 254 271 247 264 272
258 258 233                 
You can generate the values shown in the table via the command:
gnrnd4( 922178203, 5200221) in RStudio.

Answers:
   mean = 256.75904
   median = 259.0
   mode = 247, 262, 271, 273

   range: 229 to 273
    Q1 = 247
    Q3 = 267
    σ = 12.43837 (for population)
    sx = 12.51398 (for sample)


Case 4:

For the data set in Table IV, find the mean, the median, and the mode, the first and third quartiles, and the standard deviation, both treating the data set as a sample and as a population.
Table IV
21 80 50 64 85 89 33 87 113 93 29 88 47 26 96 48 34 81 66 79
36 22 72 69 124 58 93 5 85 97 98 66 59 119 60 52 98 18 70 48
82 36 74 51 82 60 60 33 106 94 68 109 85 135 3 76 55 50 82 58
67 116 82 58 76 43 17 59 32 53 74 27 76 82 5 74 55 89 145 91
74 134 37 97 76 119 95 92 43           
You can generate the values shown in the table via the command:
gnrnd4( 945818804, 3100068) in RStudio.

Answers:
   mean = 69.04494
   median = 72.0
   mode = 82

   range: 3 to 145
    Q1 = 50
    Q3 = 89
    σ = 30.39660 (for population)
    sx = 30.56882 (for sample)


Case 5:

For the data set in Table V, find the mean, the median, and the mode, the first and third quartiles, and the standard deviation, both treating the data set as a sample and as a population.
Table V
6.38.07.310.77.310.310.56.25.37.211.59.610.28.95.410.110.14.610.311.1
8.85.47.83.38.310.39.113.31.57.88.67.08.41.46.37.54.210.811.24.8
7.08.35.012.911.29.33.66.610.09.38.43.24.19.25.88.17.27.47.512.5
12.17.15.71.65.84.34.713.48.76.35.511.97.59.010.48.412.111.53.58.3
5.78.79.111.67.95.4              
You can generate the values shown in the table via the command:
gnrnd4( 1566368504, 2700075) in RStudio.

Answers:
   mean = 7.90116
   median = 8.05
   mode = 5.4, 6.3, 7.5, 8.3, 8.4, 10.3

   range: 1.4 to 13.4
    Q1 = 5.8
    Q3 = 10.1
    σ = 2.77621 (for population)
    sx = 2.79249 (for sample)
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©Roger M. Palay     Saline, MI 48176     January, 2018
The modification date for this document is 01/19/2018 03:33:42